I've have a 6-string lap steel tuned to open E at the moment and I have also used open D and C6-tuning.
I have a following problem specially with open E an D, because I cannot figure out how to build minor chords
with these tunings.
I have a following progression I tried to back up with lap steel with poor result:
Em,Am,D,G,Em,Am,D,Am,D,F,C etc.
Should I try to find common notes of each chord and use them or is there a way to build up minor chords in E and D-tuning?

let me know if it opens for you.
here is the story.... I have been playing lap steel for several years now. I tune the same manner you have. my band mate, the bass player, teaches guitar full time. he is also an excellent jazz guitarist. he knows theory backwards and frontwards. a real talent.
in my band I play several instruments each with separate tuning. two lap steels were C6 and open E. my tele is tuned open G. the band mate was always interested in how many tunings I could bring to the bands sound and repertoire.
he was curious about the lap steel having never seen one. I had an extra and lent it to him. it was tuned C6 ( CEGACE). a few hours later he emailed me. he used his theory and tweaked the tuning to give him a more user friendly fretboard.. a few hours later he sent another e mail. he said he created a new tuning that works really well. here it is: CEG BDF low to high.

strumming all the strings does not make a pleasant chord sound. but if you review the thread and peak at his videos, you will find this tuning to be quite useful. there are minor chords, seventh chords, major chords, extended chords. all separated by thirds. I can't explain it well. see the thread.
I know tune one of my gigging steels to this tuning. I can play blues, country, rock, and jazz. there are forward slants and reverse slants if you want a pedal bend sound.
try it out. you may not need another tuning.

I still use my open E. it works best with the power chord rock and Lindley's 'Mercury Blues".

hello!
Thks for yr piece of advise.
I think I have understood something for example abt. the form of moveable chords.
I try to study these pages of yr friend.
I have one more question it may be a silly one, but when you back up somebody
do you just play these chords or do you make fills out of these chord structures playing as
arpeggios or how you do your backing?
I think my biggest problem is that I cannot do nice fills.

yeah, fills. I fit little licks in between the verse. arpeggios, since I run single notes. there is no need for a full chord since the rhythm guitar has that covered. when checking out youtube videos of old county and western bands, I notice that the 'fill' guitar is playing right along with the verse. they add emotion to a word or highlight a phrase with a run of notes, using bends. then I notice that the guitar plays fills for one verse and then the pedal steel does a verse. they rarely play on top of one another. and, it seems that no fills are played during the chorus; just chords.
the trick with fills is Taste and Timing. my motto is 'get in and get out' keep them short and sweet.